My 2002 Wrangler has a simpler but similar key. It only has the RFID anti-theft part. The dealer warned me when I bought it that a new key would cost $75 in 2001 money. I tend to keep things forever so I started wondering if I could even get a new key in ten years and what it would cost.
First I bought a $3 key without the chip at Wal-Mart; it gets me in if I lock my keys in the car. Like yours, it starts the engine but it dies in a few seconds. I discovered that using the bogus key in the ignition, but holding the real key near the ignition lock for a few seconds made the computer happy. Then I could take it away and it kept running until it turned it off.
Next I found a locksmith in California on the internet who sold the chip keys for $12 or $15. His page explained how to "introduce" them to the cars' computer. It's a modestly complex sequence of turning the ignition on and off a few times at the right rate taking your cures from various lights on the instrument panel. First you demonstrate you have a real key, and then you tell it to remember your new key number. I now have four valid keys for the Jeep. I don't know how many numbers the computer can hold, but it's at least four.
The dealer probably has a fancier way to add a key and you are probably paying to use that special equipment and for labor. On the other hand it only took me 5 minutes per key to do the job manually once I found out how.
Obvious flaw in the system. If there is a receiver chip in the car that says everything is OK, its output can be clugged up to always say things are cool.

Hi. Making such cheezy stuff decide if you´re stuck is not very nice.
However, the repair will probably last as long as a new fob, if made in the same sloppy way. Regarding the cost , manufacturing such an item in the thousands should not cost more than 3-5$US a piece.
Caution: I would not recommend the solder slam method- especially with SMD pcb´s a microscopic solder splat may be too small to see and foil any further attempts to repair by destroying some circuit part when applying the battery. Better play it safe and use suction wire.
In a through hole, I usually wet the suction wire with a tiny bit of solder, the cut it to a point that will go into the Through hole and lean it up with heat. you can also clean it using a pencil graphite core (0.5mm) or stainless steel wire and gentle heat to push out the solder. In the key fob, I would have applied a little fresh solder on the top part of the solder tag, and simply pushed it down using the soldering iron.

I'd like to see a little more details of the design described. No real info on the electronics. I don't like designs with extra, unused buttons - looks sloppy, confuses users. Yes, many systems fail because of something other than electronics - like battery mounts or connectors. But look at Dodge's point of view: you had 3 good years before the quality problem showed up and then they can recoup some costs by charging an arm and leg for the replacement - which has no bearing on the BOM cost. When I do a repair such as you did I mumble to myself "it's not right, but I should get 5 more good years out of it" (I can usually improve on the original).
But PLEASE, someone do some editing. Just because spell-checking doesn't catch it doesn't mean the following words are correct: "safety, feature", When [With], batter, On half [One half], were was, indemnified [identified], devices [device], handless [handles (what is hand-less??)], work. [work?], and clean hole. There are also various comma problems. Humans are still the best. Plus Figures 3 & 4 look identical to me.

I'm not sure the flaws in design rely dolely on poor design. I have examined too this problem and found marketing reasons. Keys become unstable when it is easier to push customer towards a new car........

Nice analysis. I agree stuff like automatic alarms should be something the owner could disable.
Service tech tip. Clearing solder from a PCB through hole typically requires suction from a solder bulb or a vacuum desoldering station. A reliable field expediant is to thoroughly heat the solder joint then slam the PCB flat down on a slightly resiliant surface (like an antistatic mat on a work bench). The PCB stops when it hits the bench, but inertia pulls the molten solder out of the hole. Clean up any solder splatter from the PCB and you are good to go.
If this doesn't work the first time try adding a bit of solder to the hole. This works best if the hole is completely full of solder. Also make sure the solder is completely melted. You need to do the slam quickly before the solder gets a chance to resolidify.